Geodesic
Non-Exactness of Direct Products of Quasi-Coherent Sheaves
Documenta mathematica, Tome 24 (2019), pp. 2037-2056.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasi-projective schemes over commutative noetherian rings. The main tools of the proof are the Gabriel-Popescu embedding and Roos' characterization of Grothendieck categories satisfying Ab6 and Ab4*.
Classification : 14F05, 18E20, 16D90, 16W50, 13C60
Keywords: quasi-coherent sheaf, divisorial scheme, invertible sheaf, direct product, Gabriel-Popescu embedding, Grothendieck category 
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     author = {Kanda, Ryo},
     title = {Non-Exactness of {Direct} {Products} of {Quasi-Coherent} {Sheaves}},
     journal = {Documenta mathematica},
     pages = {2037--2056},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a15/}
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Kanda, Ryo. Non-Exactness of Direct Products of Quasi-Coherent Sheaves. Documenta mathematica, Tome 24 (2019), pp. 2037-2056. http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a15/